A Rank Theorem of Operators between Banach Spaces

Abstract

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T₀∈B(E, F) with a generalized inverse T₀⁺ ∈B(E, F). This paper shows that, for every T∈B(E, F) with T₀+(T-T₀) <1, B(I + T₀⁺(T-T₀))-¹T₀⁺ is a generalized inverse of T if and only if (I-T₀⁺T₀)N(T) = N(T₀), where N( ") stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T₀ is a semi-Fredholm operator but not in general.